I Sat Down To Work On The “Sh Is For Shalebark” Page And Got The Basic Layout And Text Done. Then

Sisters

After some thought, she decided that the concept of loving siblings was a little like the Allomantic pulse lengths she was supposed to be looking for - they were just too unfamiliar for her to understand at the moment.

Oh, Vin. Your brother may have helped you survive, but he was a horrible, horrible brother. And she HAD a baby sister that their mom killed.  I can't even...

Vin still needs to have Lift as a little sister.  Then, one night when Shallan is out on a mission as Veil she needs to come across them and bring them in off the streets and eventually be accepted into their little group as the big sister. She would try to tame them somewhat and teach them things like writing, but they would spend just as much time corrupting Shallan.  They would create an alter-ego for Lift so that the Ladies Davar, Valette and [I'm still trying to come up with a good name] could go out together, but they would also create a third alter-ego for Shallan (beacuse they wouldn't want the ghostbloods to recognize Veil) so that they could sneak out in the streets together.

Now I really want to know if burning copper affects lightweaving at all... Can Vin see through lightweaving if she is burning copper? or maybe if she is burning tin? hmmm.... Also, if Shallan weaves a disguise for another Radiant, can they maintain it with their stormlight even if they aren't a light weaver?

Rithmatics: Part 6, 9 Point Conics and Triangle Centers

Note: From this point on we are drifting farther and farther from what we know from the book. The math is all solid, but its application to Rithmatics is much more speculative.

In rithmatics, the 9-point circle plays an important role in constructing lines of warding and identifying bind points.  We also know that there exist elliptical lines of warding and that they "only have two bind points."  Now, in math we are frequently told things like "You can't take a square root of a negative number", which are true in the given system (real numbers) but not true in general.  The construction for the 9-point circle, as described in the book, doesn't work for ellipses.  However, there is a generalized 9-point conic construction.  To understand it, we need to start with a little bit of terminology.

A complete quadrangle is a collection of 4 points and the 6 lines that can be formed from them.  For our purposes, we will be concerned with complete quadrangles formed from the vertices of the triangle and a point inside the triangle.  The 6 lines are then the sides of the triangles and the three lines connecting the center point to the vertices.

The diagonal points of a complete quadrangle are the three intersection points formed by extending opposite sides of the quadrangle.  If we have a triangle ABC with center P, then the intersection of AB with PC is a  diagonal point.

If you take the midpoints of the 6 sides of a complete quadrangle and the 3 diagonal points of that quadrangle, these 9 points will always lie on a conic. This conic is the 9-point conic associated with the complete quadrangle.

Note that if we choose our point in the center of the triangle to be the point where the altitudes meet (known as the orthocenter), then this construction is exactly what we have been doing to create 9-point circles.

There are four classical and easily constructable triangle centers - the orthocenter, circumcenter, centroid, and the incenter.  There are over 5000 other possible notions of the center of a triangle, but most of them cannot be easily geometrically constructed and they get increasingly complicated. 

Let's look at each of these 4 triangle centers and the conic they produce for a particular triangle. We will use a 40-60-80 triangle in each case for illustration purposes, but the results will be very similar for any acute triangle with 3 distinct angles.

Orthocenter: We already know about the orthocenter (that is what most of this series has been focused on so far).  For reference, here is what the 9-point circle for this triangle looks like:

Circumcenter: The circumcenter of a triangle is found by finding the midpoint of each side of the triangle and drawing in the perpendicular bisectors.  The points where the perpendicular bisectors meet is the circumcenter.  Note: This point is also the center of the circle that can be circumscribed around the triangle.

Unlike with the orthocenter, the lines we use to construct the circumcenter (the dashed lines in the diagram) are not part of the complete quadrangle, so we have to finish the quadrangle after we have identified the circumcenter.  The resulting conic is an ellipse.

Centroid: The centroid of a triangle is formed by finding the midpoint of each side of the triangle and connecting it to the opposite vertex.  The intersection of these median lines is the centroid.

The lines used to construct the centroid are part of the complete quadrangle, but we have the interesting situation where the centers of each side are also the diagonal points of the complete quadrangle.  This means that, regardless of the triangle used, we will only ever have 6 distinct points.  The resulting conic is an ellipse that is tangent to all three sides of the triangle.

Incenter: The incenter of a triangle is the intersection of the  angle bisectors of the triangle.  

Note that the lines used to construct the incenter of the triangle are also the additional lines of the complete quadrangle.  In addition, as long as the angles of the original triangle are distinct, the 9 points in the construction will all be distinct.  The resulting conic is an ellipse.

In Summary:  There are lots of ways that we could potentially construct a 9-point ellipse from a triangle.  Of these options, I would guess that the construction using the  incenter of the triangle is the most likely to produce valid rithmatic structures.  I lean this way because, as with the orthocenter, constructing the incenter also constructs the complete quadrangle and its diagonal points.  Furthermore, the 9 points of the construction will all be distinct (except in special cases). As such, we will explore 9-point ellipses constructed with the incenter more thoroughly in the next post. 

Ok, so after the skyeel came out yesterday, I decided to try a chull today.  I'm not entirely happy with it, but here, have it anyway.

Ok.  I sat down planning to draw a Sarpenthyn.  You know, the critter in the menagerie that everyone but Shallan though was disgusting.  And then this adorable creature came out and I’m so confused. What do y’all think? Could this be a  sarpenthyn?

Rithmatics: Part 1

Hello!  I'm rather fascinated with Rithmatics (the magic system in Brandon Sanderson's The Rithmatist) at the moment, which means you are going to be getting a series of mathy posts.  They will all be tagged with #rithmatics .  I've been encouraged to add the cfsbf tag.  If this bothers anyone, please let me know.

In this first post we explore how all of the binding patterns for circular defenses can be derived from 9 point circles. You might be able to get everything from the pictures, but I give explanations as well.

9 Point Defenses

Let's start by talking about 9 point circles.  Start with a triangle.  Absolutely any triangle will do, but for 9 point defenses we want acute triangles (all angles less than 90 degrees) where all of the angles are distinct.  Mark the midpoints of each side and draw in the three altitudes (start at each vertex and draw the line perpendicular to the opposite side) of the triangles.  Mark the points where the altitudes intersect the sides of the triangles (there are 3 such points, one for each altitude).  Note that all of the altitudes meet a single point. Mark the midpoint of each segment connecting P to one of the vertices of the triangle.  This gives you three more points for a total of 9.  These 9 points will be distinct and lay on a circle.

This explains how to get the bind points for any 9-point defense.  However, not all defenses have 9 points.  These turn out to be very special cases of 9 point triangles where some of the points coincide.  

6 Point Defenses

To get 6 points, start with an equilateral triangle.  Any time two angles of a triangle have the same measure, the altitude from the third angle will bisect its opposite side.  Since all of the angles are the same here, all of the altitudes bisect their opposing sides.  This gives us a "9-point" circle with 6 evenly spaced points.

4 Point Defenses

This time we want an isosceles right triangle (you might know it better as a 45-45-90 triangle).  In right triangles, the legs are also altitudes, which means that the vertex at the right angle is also the point where the altitudes intersect each other. It is also the point where each leg "intersects" the other and the "half way point" between the intersection of the altitudes and itself, so it counts as 3 of the 9 points.  The resulting 4 points form a square and so are evenly spaced around the circle.

2 Point Defenses

This is the strangest case.  Here our triangle is degenerate - one of the sides has length 0, which means that the "triangle" is just a line.  To see how to follow the 9 point construction in this case, we can look at a limit.  Start with a really skinny isosceles triangle.  If you follow the construction, you get three points grouped near each approximately half way up the triangle.  The other 6 points are clustered down near the narrow base.  Now pretend the narrow point is a hinge and slowly close it. As you do, the three points in the middle get closer and closer together, the base gets narrower and narrower and the 6 points near it get closer and closer together.  In the limit this gives us a line segment and a circle which uses half of the line segment as a diameter

I used fallenwithstyle's chart to knit myself a Pattern hat! The brown is KnitPicks City Tweed DK and the purple is KnitPicks Wool of the Andes sport. They were both sitting in my stash of yarn waiting to be right for something. I need to work on my consistency with keeping the floats loose enough in stranded knitting, but it isn't bad enough to be a problem.The hat also came out slightly longer than I usually prefer, but I'm super happy with it anyway :-). While I was knitting it, it occurred to me that there is potential for an amazing Pattern hat done with cables rather than colorwork. I'm not sure when I will get to it, but designing that pattern has gone in my "projects I will get to at some point" file. 

Sprenhead!

Here’s my finished Pattern hat! I’m really happy with how it came out not just as fanart but as a hat in general (and I said I didn’t need any more hats…).

I changed the chart sightly from the original version I posted, so here is the final version (it doesn’t include instructions for the flared brim or picot hem though, just the colorwork). 

Oathbringer Speculation: Timbre

The descriptions of Timbre would fit with the name “lightspren”. We know that she communicates with Venli by pulsing to different rhythms. During the first shadesmar boat trip, Shallan speculates that the Reachers/lightspresn are using vibrations (ie pulses) to communicate.

This suggests that Timbre is a Reacher, but I suspect that she’s not just any Reacher. We meet Ico in Shadesmar and learn that his father is a deadeye and his daughter “ran off chasing stupid dreams”. Then Timbre tells Venli that her own grandfather was lost to human betrayal. Putting all of this together, I strongly suspect that Timbre is Ico’s daughter. I’m not sure what implications this will have for the future. But it’s a thing.

Citations: (Note: Page numbers come from the Kindle edition.)

What was that small spren that had crept out from beneath Eshonai’s corpse? It looked like a small ball of white fire; it gave off little rings of light and trailed a streak behind it. Like a comet. (pg 340) 

“The copper vibrates,” Shallan said. “And they keep touching it. I think they might be using it to communicate somehow.” (pg 931)

 “Wait!” Adolin said. “Ico, I saw something moving back there.” Ico locked the door and hung the keys on his belt. “My father.” “Your father?” Adolin said. “You keep your father locked up?” “Can’t stand the thought of him wandering around somewhere,” Ico said, eyes forward. (Pg 946)

Ico speaking:  “My daughter used to work there, before she ran off chasing stupid dreams.”  (Pg 948)

Timbre pulsed to Irritation, then the Lost. “That many? I had no idea the human betrayal had cost so many of your people’s lives. And your own grandfather?” (pg 1196)

Happy Pi Day! Have some chocolate π